Доступ предоставлен для: Guest
Портал Begell Электронная Бибилиотека e-Книги Журналы Справочники и Сборники статей Коллекции
International Journal for Uncertainty Quantification
Импакт фактор: 3.259 5-летний Импакт фактор: 2.547 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN Печать: 2152-5080
ISSN Онлайн: 2152-5099

Свободный доступ

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2018021551
pages 61-73

A MULTI-INDEX MARKOV CHAIN MONTE CARLO METHOD

Ajay Jasra
Department of Statistics & Applied Probability, National University of Singapore, Singapore
Kengo Kamatani
Graduate School of Engineering Science, Osaka University, Osaka, 565–0871, Japan
Kody J. H. Law
School of Mathematics, University of Manchester, Manchester, UK, M13 9PL
Yan Zhou
Department of Statistics & Applied Probability National University of Singapore, Singapore

Краткое описание

In this paper, we consider computing expectations with respect to probability laws associated with a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in space-time. In such contexts, it is known that the multi-index Monte Carlo (MIMC) method of Haji-Ali, Nobile, and Tempone, (Numer. Math., 132, pp. 767– 806, 2016) can improve on independent identically distributed (i.i.d.) sampling from the most accurate approximation of the probability law. Through a nontrivial modification of the multilevel Monte Carlo (MLMC) method, this method can reduce the work to obtain a given level of error, relative to i.i.d. sampling and even to MLMC. In this paper, we consider the case when such probability laws are too complex to be sampled independently, for example a Bayesian inverse problem where evaluation of the likelihood requires solution of a partial differential equation model, which needs to be approximated at finite resolution. We develop a modification of the MIMC method, which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem for a simplified estimator that shows that using our MIMCMC method is preferable, in the sense above, to i.i.d. sampling from the most accurate approximation, under appropriate assumptions. The method is numerically illustrated on a Bayesian inverse problem associated to a stochastic partial differential equation, where the path measure is conditioned on some observations.


Articles with similar content:

FORWARD AND INVERSE UNCERTAINTY QUANTIFICATION USING MULTILEVEL MONTE CARLO ALGORITHMS FOR AN ELLIPTIC NONLOCAL EQUATION
International Journal for Uncertainty Quantification, Vol.6, 2016, issue 6
Ajay Jasra, Yan Zhou, Kody J. H. Law
SURROGATE MODELING FOR STOCHASTIC DYNAMICAL SYSTEMS BY COMBINING NONLINEAR AUTOREGRESSIVE WITH EXOGENOUS INPUT MODELS AND POLYNOMIAL CHAOS EXPANSIONS
International Journal for Uncertainty Quantification, Vol.6, 2016, issue 4
Eleni N. Chatzi, Minas D. Spiridonakos, Chu V. Mai, Bruno Sudret
ROBUST ADAPTIVE CONTROL OF SISO DYNAMIC HYBRID SYSTEMS
Hybrid Methods in Engineering, Vol.2, 2000, issue 1
M. de la Sen
Convergence of a Matrix Gradient Control Algorithm with Feedback Under Constraints
Journal of Automation and Information Sciences, Vol.32, 2000, issue 10
Yarema I. Zyelyk
A WEIGHT-BOUNDED IMPORTANCE SAMPLING METHOD FOR VARIANCE REDUCTION
International Journal for Uncertainty Quantification, Vol.9, 2019, issue 3
Linjun Lu, Tenchao Yu, Jinglai Li